Existence of Positive Solutions to Nonlinear Fractional Boundary Value Problem with Changing Sign Nonlinearity and Advanced Arguments

We discuss the existence of positive solutions to a class of fractional boundary value problem with changing sign nonlinearity and advanced arguments 𝐷 𝛼 𝑥(𝑡) + 𝜇ℎ(𝑡)𝑓(𝑥(𝑎(𝑡))) = 0, 𝑡 ∈ (0, 1),2 < 𝛼 ≤ 3,𝜇 > 0, 𝑥(0) = 𝑥 󸀠 (0) = 0,𝑥(1) = 𝛽𝑥(𝜂) + 𝜆[𝑥],𝛽 > 0, and 𝜂 ∈ ( 0 , 1 ), where 𝐷 𝛼 is the standard Riemann-Liouville derivative, 𝑓 : [0,∞) → [0, ∞) is continuous, 𝑓(0) > 0 , ℎ : [0,1] → (−∞,+∞) , and 𝑎(𝑡) is the advanced argument. Our analysis relies on a nonlinear alternative of Leray-Schauder type. An example is given to illustrate our results.


Introduction
Fractional differential equations (FDEs) have been of great interest for the past three decades.It is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in the modeling of many phenomena in various fields of science and engineering.Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, and so forth (see [1,2]).Therefore, the theory of FDEs has been developed very quickly.There has been a significant development in fractional differential equations in recent years; see .

Basic Definitions and Preliminaries
In this section, we present some preliminaries and lemmas that are useful to the proof of our main results.For convenience, we also present the necessary definitions from fractional calculus theory here.These definitions can be found in the recent literature.
Definition 1.The fractional integral of order  > 0 of a function  : (0, +∞) →  is given by provided that the right-hand side is pointwise defined on (0, +∞).
Definition 2. The fractional derivative of order  > 0 of a continuous function  : (0, +∞) →  is given by where  = []+1 and [] denotes the integral part of number , provided that the right-hand side is pointwise defined on (0, +∞).
Lemma 4. Assume that  −1 ̸ = 1 and  ∈ (,); then problem (7) has the unique solution given by the following formula: where Theorem 5. Let  be a Banach space with  ⊂  closed and convex.Assume that  is a relatively open subset of  with 0 ∈  and  :  →  is a continuous, compact map.Then either (i)  has a fixed point in  or (ii) there exist  ∈  and  ∈ (0, 1) with  = .

Existence of Positive Solutions
Let us denote by  = [0,1] the Banach space of all continuous real functions on [0, 1] endowed with the sup norm and let  be the cone: Lemma 6.Let assumptions (H 1 )-(H 4 ) hold.Moreover, we assume that assumptions (H 5)-(H 6 ) hold with where Then, for every 0 <  < 1, there exists a positive number  such that, for 0 <  < , the nonlinear fractional differential equation, has a positive solution   with ‖  ‖ → 0 as  → 0 and where Proof.It is easy to know from ( 9), (H 5 ), and (H 6 ) that () > 0,  ∈ (0, 1].By Lemma 4, ( 12) has a unique solution in : For  ∈ (,  + ), we define two operators  and  by where It is easy to show that  :  →  and  :  →  are completely continuous.We claim that operators  and  have the same fixed points in .In fact, let  = ; then So This shows that fixed points of  are solutions of (12).We will apply the nonlinear alternative of Leray-Schauder type to prove that  has at least one fixed point for small .